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Econometrica, Vol. 80, No. 2 (March, 2012), 631–660
HIGHER ORDER UNCERTAINTY AND INFORMATION:
STATIC AND DYNAMIC GAMES
ANTONIO PENTA
University of Wisconsin—Madison, Madison, WI 53706, U.S.A.
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Econometrica, Vol. 80, No. 2 (March, 2012), 631–660
HIGHER ORDER UNCERTAINTY AND INFORMATION:
STATIC AND DYNAMIC GAMES
BY ANTONIO PENTA1
Weinstein and Yildiz (2007) have shown that in static games, only very weak predictions are robust to perturbations of higher order beliefs. These predictions are precisely those provided by interim correlated rationalizability (ICR). This negative result
is obtained under the assumption that agents have no information on payoffs. This assumption is unnatural in many settings. It is therefore natural to ask whether Weinstein
and Yildiz’s results remain true under more general information structures. This paper characterizes the “robust predictions” in static and dynamic games, under arbitrary
information structures. This characterization is provided by an extensive form solution
concept: interim sequential rationalizability (ISR). In static games, ISR coincides with
ICR and does not depend on the assumptions on agents’ information. Hence the “no
information” assumption entails no loss of generality in these settings. This is not the
case in dynamic games, where ISR refines ICR and depends on the details of the information structure. In these settings, the robust predictions depend on the assumptions
on agents’ information. This reveals a hitherto neglected interaction between information and higher order uncertainty, raising novel questions of robustness.
KEYWORDS: Dynamic games, hierarchies of beliefs, higher order beliefs, information, interim sequential rationalizability, robustness, uniqueness.
1. INTRODUCTION
ECONOMIC MODELLING TYPICALLY INVOLVES making common knowledge assumptions. It is therefore natural to ask which predictions of a model retain
their validity when such assumptions are relaxed. Recently, Weinstein and
Yildiz (2007) have shown that in static games, such predictions are precisely
those provided by interim correlated rationalizability (ICR; Dekel, Fudenberg,
and Morris (2007)). They showed that any ICR action profile can be made
uniquely ICR by perturbing agents’ higher order beliefs (their structure theorem). This implies that any refinement of ICR is not robust, as it rules out outcomes that would be uniquely selected in some arbitrarily close model. Overall,
the result has important negative implications: no equilibrium concept delivers
robust predictions once common knowledge assumptions are relaxed.
To prove the structure theorem, Weinstein and Yildiz (2007) assumed a richness condition on the underlying space of uncertainty. This condition requires
1
Previous versions of this manuscript circulated under the title “Higher Order Beliefs in Dynamic Environments.” I want to thank my advisor, George Mailath, for his dedication and for
the encouragement, much needed at some stages of this project. I am grateful for the helpful
comments of a co-editor and three anonymous referees, as well as Yi-Chun Chen, Eddie Dekel,
Amanda Friedenberg, Qingmin Liu, Stephen Morris, Andrew Postlewaite, Marzena Rostek, and
seminar and conference audiences at UPenn, Bocconi, 2008-NASM of the ES, 2008-ESEM, and
IX-SAET Conference. Special thanks go to Pierpaolo Battigalli, to whom I am particularly indebted.
© 2012 The Econometric Society
DOI: 10.3982/ECTA9159
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ANTONIO PENTA
that for every action of every player there exists a state under which that action is strictly dominant. This means that essentially all common knowledge
assumptions are relaxed, as under the richness condition it is not common
knowledge among the players that any action is not dominant. Weinstein and
Yildiz also assumed that players have no information about payoffs (their own
or their opponents’) and that this is common knowledge.
The “no information” assumption is unnatural in many settings. For example, agents competing in an auction may have private information on the value
of the object; in bilateral trade environments, it may be natural to assume that
traders know their valuation for the good, and sometimes one may wish to
maintain that this is common knowledge too (i.e., assume private values). In
yet other settings, one may not wish to impose common knowledge that agents
have no information, allowing instead agents to be uncertain as to whether
their opponents are privately informed or not. For instance, in an auction it
may be common knowledge that some agents have information and not others, but the identities of the informed and uninformed may not be common
knowledge. It is natural then to ask whether the negative results of Weinstein
and Yildiz are specific to the no information assumption, and to investigate
what predictions are robust under more general information structures.2
In this paper, I characterize the predictions that are robust to the relaxation
of common knowledge assumptions, in both static and dynamic games, when
higher order beliefs are perturbed in general information structures. To this
end, I introduce a new solution concept: interim sequential rationalizability
(ISR). ISR is an extensive form solution concept, which coincides with ICR in
static games. I show that, irrespective of the information structure, a structure
theorem analogous to Weinstein and Yildiz’s holds for ISR.
In static games, ISR does not depend on the assumptions on the information structure. Hence, Weinstein and Yildiz’s results extend to arbitrary information structures: the no information assumption entails no loss of generality.
But this is not true in dynamic games, as ISR depends on the assumptions on
agents’ information, raising novel questions of robustness.
These results show that, while unimportant in static games, the fine details
of informational assumptions may be crucial in dynamic games. This has important implications for applied theory. For instance, Bergemann and Morris
(2007) recently argued that in auction problems with interdependent values,
ascending auctions can be useful in reducing strategic uncertainty so as to eliminate undesirable outcomes that could arise in their sealed-bid counterparts.
2
A complementary question would be to consider less demanding robustness tests, in which
some common knowledge assumptions are maintained (for instance, that some actions are not
dominant). In Penta (2011), I studied the structure of ICR on arbitrary spaces of uncertainty,
that is, without assuming richness. I showed that the structure theorem remains true under very
mild relaxations of the common knowledge assumptions, which reinforces Weinstein and Yildiz’s
message.
HIGHER ORDER UNCERTAINTY AND INFORMATION
633
The results of the present paper suggest that the robustness of that insight
depends on the details of agents’ information about payoffs: for instance, in
perturbing agents’ higher order beliefs, if we assume that agents have no information about payoffs, all the advantages of the ascending auction disappear.
On the contrary, the robustness properties of sealed-bid auctions are not affected by changes in the informational assumptions.
More generally, it can be argued that the no information assumption is too
restrictive in dynamic games.3 The main reason for studying these games is
that if agents cannot commit to their strategies, “sequential rationality” considerations impose stronger restrictions than those implied by the normal form
approach. But once common knowledge assumptions are relaxed, the very notion of sequential rationality has no bite if agents have no information on payoffs. This is because, with no information, players’ beliefs about payoffs after
unexpected moves are only restricted by what is commonly known. If all such
assumptions are relaxed (e.g., assuming richness), sequential rationality has no
bite at zero probability histories, which means that it coincides with “normal
form rationality.” Notice that this reasoning does not involve higher order beliefs. It follows immediately from the richness and no information assumptions
combined.
In seeking to characterize the robust predictions in dynamic games, it is natural to look at extensive form analogues of ICR, that is, dynamic counterparts of
the assumptions of rationality and common belief in rationality. The weakest of
such counterparts assumes that agents are sequentially rational and that this is
common belief at the beginning of the game (initial common belief in sequential
rationality (ICBSR)). A natural question, therefore, is how to characterize the
“strongest robust predictions” that satisfy such a minimal requirement. ISR
comprises precisely such a strongest robust solution concept: refinements of
ISR (e.g., extensive form rationalizability or perfect Bayesian equilibrium) are
not robust. On the other hand, ISR characterizes the behavioral implications
of ICBSR. Hence, the strongest robust predictions are also the weakest among
those consistent with ICBSR.
Weinstein and Yildiz (2007) also proved that once common knowledge assumptions are relaxed, static games are generically dominance-solvable. This
result generalizes an important insight from the literature on (static) global
games: the pervasive multiplicity of equilibria that we observe in standard models stems from the high degree of coordination implicit in the common knowledge assumptions. The validity of this insight in dynamic contexts has been
questioned by the recent literature on dynamic global games, in which the familiar uniqueness results do not obtain. By proving a generic uniqueness result
3
Independent work by Chen (2011) extended the structure theorem for ICR to (finite) dynamic
games in normal forms, maintaining the no information assumption. The relationship with his
work is discussed in Section 4.1.
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ANTONIO PENTA
for ISR, I show in contrast that the same insight extends to dynamic games,
irrespective of the information structure.4
The rest of the paper is organized as follows. Section 2 introduces the game
theoretic framework. Section 3 introduces ISR and some of its properties. Section 4 provides the structure theorem. Section 5 introduces a novel notion of
robustness (information invariance) motivated by the role that assumptions on
information have in dynamic games. Section 6 concludes.
2. GAME THEORETIC FRAMEWORK
The analysis that follows applies to multistage games with observable actions
(Fudenberg and Tirole (1991, Sect. 8.2)),5 which are defined by an extensive
form that represents agents’ possible moves and information about the opponents’ moves, a preference-information structure (PI structure hereafter) that
represents players’ information about everyone’s payoffs, and a type space that
represents agents’ beliefs.6 These concept are formally introduced next.
2.1. Extensive Forms
An extensive form is defined by a tuple
Γ = N H Z (Ai )i∈N where N = {1 n} is the set of players and for each player i, Ai is the (finite)
set of his possible actions. Histories are finite concatenations of action profiles,
and the (finite) set of all possible histories is partitioned into the set of terminal
histories Z and the set of partial histories H (the latter includes the empty
history φ). As the game unfolds, the partial history h that has just occurred
becomes public information and is perfectly recalled by all players. For each
h ∈ H and i ∈ N, let Ai (h) denote the (finite) set of actions available to player i
at history h, and let A(h) = i∈N Ai (h) and A−i (h) = j∈N\{i} Aj (h).7 Without
loss of generality, Ai (h) is assumed to be nonempty for each h: player i is
inactive at h if |Ai (h)| = 1 and he is active otherwise. This setup allows finitely
×
×
4
Morris and Shin (2003) surveyed the literature on (static) global games. On dynamic global
games, see, for instance, Angeletos, Hellwig, and Pavan (2007), who considered an infinite horizon game, which does not fall within the present setting. Nonetheless, as they discussed (Angeletos, Hellwig, and Pavan (2007, p. 729)), their multiplicity result also holds in the finite horizon
version of their model.
5
The analysis can be easily extended to all finite dynamic games with perfect recall, but restricting attention to multistage games with observable actions significantly simplifies the notation.
6
This terminology is not entirely standard: Bayesian games are usually defined by an extensive
form, payoff functions, and a type space. Information and beliefs are often conflated in the type
space. The separation proposed here is common in the literature on robust mechanism design
(see references in footnote 11).
7
Formally, Ai (h) = {ai ∈ Ai : ∃a−i ∈ A−i s.t. (h (ai a−i )) ∈ H ∪ Z }.
HIGHER ORDER UNCERTAINTY AND INFORMATION
635
repeated games as a special case or games with perfect information if H is such
that only one player is active at each h. If H = {φ}, the game is static.
Pure strategies of player i assign to each partial history h ∈ H an action in
Ai (h). Let Si denote the set of reduced form strategies (or plans of action) of
player i. Two strategies correspond to the same reduced strategy si ∈ Si if and
only if they are realization-equivalent to si , that is, they preclude the same
collection of histories and for every nonprecluded history h, they select the
same action, si (h).8 Each profile s induces a unique terminal history z(s) ∈ Z .
For each h ∈ H, let Si (h) be the set of strategies si that allow h to be reached
(that is, there exists s−i such that h is on the path to z(si s−i )). Let H(si ) =
{h ∈ H : si ∈ Si (h)} denote the set of partial histories not precluded by si . Since
only reduced strategies are considered here, I omit the term “reduced” in the
following discussion.
2.2. PI Structures
Players’ payoffs are represented by functions ui : Z → R for each i. To model
situations in which payoffs are not common knowledge, payoff functions are
parametrized on a space Θ, ui : Z × Θ → R.9 Elements of Θ are referred to
as payoff states. Players’ information is modelled as an information partition
on Θ. To avoid unnecessary complications, I focus on partitions with a product
structure, so that Θ can be written as10
Θ = Θ0 × Θ1 × · · · × Θn For each i = 1 n, Θi is the set of player i’s payoff types. Θ0 is referred
to as the set of states of nature. Sets Θk (k = 0 1 n) are assumed to be
compact subsets of Euclidean spaces and each ui is concave in θi . Also define
Θ−i = j∈N\{i} Θj , so that Θ = Θ0 × Θi × Θ−i .
In state (θ0 θ1 θn ), player i’s payoff type is θi , which is observed at
the beginning of the game. Hence, payoff types represent agents’ information
about the payoff state: if i’s payoff type is θ̂i , i knows that the true state belongs
to the set Θ0 × {θ̂i } × Θ−i . The set Θ0 represents the residual uncertainty that
is left after pooling everybody’s information.
×
8
The standard notion of strategy specifies a player’s behavior also at histories precluded by
the strategy itself. In equilibrium, this counterfactual behavior represents the opponents’ beliefs
about i in case he has deviated from the strategy. This paper follows a nonequilibrium approach,
and players’ beliefs about the opponents’ behavior are modelled explicitly. We can thus restrict
attention to plans of actions.
9
This representation is without loss of generality. For example, taking Θ ≡ ([0 1]n )Z and letting u(z θ) = θ(z) imposes no restrictions on agents’ payoffs.
10
The extension to general information partitions is conceptually straightforward, but notationally cumbersome. The same simplifying restriction is common in the literature on mechanism
design (e.g., references in footnote 11).
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ANTONIO PENTA
The tuple Θ0 (Θi ui )i∈N thus represents agents’ information about payoffs. It is assumed to be common knowledge and is referred to as preferenceinformation structure (PI structure). Here are a few examples of special cases:
(i) If Θk is a singleton for all k = 0 1 n, then the game has complete
information.
(ii) If Θ0 (Θi ui )i∈N is such that, for every i ∈ N, ui is constant in (θ0 θ−i ),
the PI structure is one of private values.
(iii) If Θi is a singleton for every i (hence Θ Θ0 ), then agents’ have no
information about payoffs and this is common knowledge. This is the special
case considered by Weinstein and Yildiz (2007) and Chen (2011). (Notice that
private values is neither a stronger nor a weaker assumption than no information.)
(iv) If Θ0 is a singleton, the PI structure is one of distributed knowledge.
These structures are common in the literature on robust mechanism design.11
(v) The private values and no information assumptions are somewhat extreme and sometimes too restrictive. For instance, it may be desirable to allow
agents to have private information without necessarily imposing that this is
common knowledge. For example, let Θ0 = [1 2], Θi = [0 3], and payoff functions ui be such that for some Ui ∈ [0 1]Z , ui (z θ) = (θi − θ0 )Ui (z). Hence, if
θi > 2 or θi < 1, player i knows his own preferences (Ui and −Ui , respectively),
but if θi ∈ [1 2], player i has essentially no information. In such a PI structure,
whether i is informed is not common knowledge (of course, other common
knowledge assumptions are implicit in this example).
Information and Beliefs
Agents also entertain (subjective) beliefs about the components of the state
they do not know. For instance, agent i may assign probability 1 to some strict
subset E ⊂ Θ, that is, he is certain of E. Beliefs are introduced in Section 2.3,
but it is useful to point out an important difference between information and
beliefs (or between knowledge and certainty). First of all, while agents may
entertain wrong beliefs (i.e., be certain of false events), information is never
false in this model. Player i knows θ̂i only if the realized state is in Θ0 × {θ̂i } ×
Θ−i , that is, if θ̂i is true.12
As the game unfolds, agents may change their beliefs about the payoff state
in response to observing the opponents’ past moves. When both information
and beliefs are represented in the same model, it is standard to maintain that
11
See, for example, Bergemann and Morris (2005, 2009) and Artemov, Kunimoto, and Serrano
(2011) for static mechanisms; see Mueller (2010) and Penta (2010a) for dynamic mechanisms.
12
The difference between knowledge and belief is well understood in epistemological game
theory. At a formal level, the difference is the so-called axiom of truth (or axiom of knowledge),
which is satisfied in knowledge structures, not in belief structures. The axiom of truth represents
precisely the idea that “i knows E” only if “E is true whenever i knows E.” (See Osborne and
Rubinstein (1994, Chap. 5).)
HIGHER ORDER UNCERTAINTY AND INFORMATION
637
agents’ beliefs never contradict their information.13 This implies that if i’s payoff type is θi , player i’s beliefs are concentrated on Θ0 × {θi } × Θ−i at every
history. That is, i always is certain of θi . In contrast, initial certainty about elements other than θi does not imply that such beliefs are maintained later in
the game: a player who initially is certain that (θ0 θ−i ) ∈ E may abandon this
belief after observing an unexpected event, such as a zero probability history.
Endowing agents with more information therefore restricts the set of feasible
beliefs after unexpected events. This is why assumptions on information, which
are inconsequential in static games, are crucial in dynamic games.
2.3. Exogenous Beliefs
To complete the description of the strategic situation, players’ beliefs about
what they do not know must be specified. That is, for every i, his beliefs about
Θ0 × Θ−i (first-order beliefs), his beliefs about Θ0 × Θ−i and the opponents’
first-order beliefs (second-order beliefs), and so on.
Given a PI structure Θ0 (Θi ui )i∈N , players’ hierarchies of beliefs are defined as usual (see Mertens and Zamir (1985)): for each i ∈ N, let Zi1 =
Δ(Θ0 × Θ−i ) denote the set of player i’s first-order beliefs, and for k ≥ 1, define
recursively14
k
Z−i
=
×Z
j=i
k
j
and
Zik+1
k
= (πi1 πik+1 ) ∈ Zik × Δ(Θ0 × Θ−i × Z−i
):
marg
k
Θ0 ×Θ−i ×Z−i
πik+1 = πik Agent i’s first-order beliefs are elements of Δ(Θ0 × Θ−i ); an element of Δ(Θ0 ×
k−1
Θ−i × Z−i
) is a Θ-based k-order belief for every k > 1. The set of (collectively
coherent) Θ-hierarchies is defined as
1
2
k−1
Δ(Θ0 × Θ−i × Z−i ) :
HiΘ = (πi πi ) ∈
×
k≥1
(πi1 πik ) ∈ Zik ∀k ≥ 1 13
This requirement is both standard and natural. The very notion of information would be
problematic if agents’ beliefs were not required to be consistent with it. In the epistemic literature,
this requirement is one of the axioms that define information in dynamic models of knowledge
and belief (see Battigalli and Bonanno (1999) for a survey).
14
For any set X, Δ(X) denotes the set of probability distributions over X, endowed with the
topology of weak convergence.
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ANTONIO PENTA
Players’ Θ-hierarchies are represented by means of type spaces.
DEFINITION 1—Θ-Based Type Space: A (Θ-based) type space is a tuple
T = Θ0 (Θi Ti θi τi )i∈N such that for each i ∈ N, Ti is a set of types, θi : Ti → Θi is an onto function that
assigns to each type a payoff type and τi : Ti → Δ(Θ0 × T−i ) assigns to each type
a belief about the states of nature and the opponents’ types.15 We focus here
on compact type spaces, in which sets Ti are compact, and functions τi and θi
are continuous.
Each type in a type space induces a Θ-hierarchy: the first-order beliefs induced by ti ∈ Ti are obtained by the map π̂i1 : Ti → Δ(Θ0 × Θ−i ) defined as
follows: for every measurable E ⊆ Θ0 × Θ−i ,
π̂i1 (ti )[E] = τi (ti ) {(θ0 t−i ) ∈ Θ0 × T−i : (θ0 θ−i (t−i )) ∈ E} For k > 1, the induced k-order beliefs are obtained by mappings π̂ik : Ti →
k−1
), defined recursively as follows: for every measurable E ⊆
Δ(Θ0 × Θ−i × Z−i
k−1
Θ0 × Θ−i × Z−i ,
π̂ik (ti )[E] = τi (ti ) {(θ0 t−i ) ∈ Θ0 × T−i : (θ0 θ−i (t−i ) π̂ik−1 (ti )) ∈ E} The map π̂i : Ti → HiΘ , defined as
ti → π̂i (ti ) = (π̂i1 (ti ) π̂i2 (ti ) )
assigns to each type in a Θ-based type space the corresponding Θ-hierarchy of
beliefs.
From Mertens and Zamir (1985), we know that when HΘ is endowed with
the product topology, there is a homeomorphism
φi : HiΘ −→ Δ(Θ0 × Θ−i × H−iΘ )
that preserves beliefs of all orders: for each πi = (πi1 πi2 ) ∈ HiΘ ,
marg
Θ0 ×Θ−i ×Zik−1
φi (πi ) = πik
∀k ≥ 1
15
The requirement that θi : Ti → Θi be onto is not essential to the results, but is conceptual:
if θi (Ti ) Θi , a type space de facto imposes common knowledge (CK) restrictions on payoffs
beyond those entailed by the PI structure. Setting Θi = θi (Ti ) guarantees that all CK assumptions on payoffs are represented by the PI structure, so that the type space only imposes extra
assumptions on beliefs. This separation is useful in defining the notion of information invariance
(Section 5). Without the “onto” requirement, that notion would be more involved. (In that case,
the notion of embedding, Definition 6, would refer to the images of the θi functions instead of
the sets Θi .)
HIGHER ORDER UNCERTAINTY AND INFORMATION
639
∗
∗
θ∗i τi∗ )i∈N , where TiΘ
:= Θi × HiΘ and for evHence, the tuple TΘ∗ = Θ (TiΘ
∗
∗
∗
ery ti = (θi πi ) ∈ TiΘ , τi (θi πi ) = φi (πi ) and θi (θi πi ) = θi , is a type space. It
is referred to as the (Θ-based) universal type space. Let π̂i0 (ti ) ≡ θi (ti ) and de∗
fine π̂i∗ : Ti → TiΘ
so that π̂i∗ (ti ) = (π̂i0 (ti ) π̂i (ti )). Mertens and Zamir showed
that for any nonredundant type space, the set π̂ ∗ (T ) is a belief-closed subset of T ∗ , in the sense that for every π̂i (ti ) ∈ π̂i (Ti ), we have φi (π̂i (ti ))[Θ0 ×
∗
∗
(T−i )] = 1.16 A finite type is any element ti ∈ TiΘ
that belongs to a finite
π̂−i
∗
belief-closed subset of TiΘ . The set of finite types is denoted by T̂iΘ .
Players’ hierarchies of beliefs (or types) are envisioned as purely subjective
states describing a player’s view of the strategic situation. As such, they enter
the analysis as a datum and are regarded in isolation (i.e., player by player and
type by type). It is given such (exogenous) beliefs that we can apply game theoretic reasoning to make predictions about players’ behavior (the endogenous
variables).
2.4. Bayesian Games in Extensive Form
Given a PI structure Θ (ui )i∈N and a Θ-based type space T , let ûi : Z ×
Θ0 × T → R be such that for each (z θ0 t) ∈ Z × Θ0 × T , ûi (z θ0 t) =
ui (z θ0 θ(t)). Function ûi extends ui ’s domain to the payoff irrelevant higher
order beliefs. To avoid unnecessary notation, in the following discussion we use
ui to denote both payoff functions.
A tuple Θ T (ui )i∈N and an extensive form Γ define a Bayesian game in
extensive form:
Γ T = N H Z Θ (Ti θi τi ui )i∈N (Notice that game Γ T need not be consistent with a common prior.17 )
3. INTERIM SEQUENTIAL RATIONALIZABILITY
Interim sequential rationalizability (ISR) is a solution concept for Bayesian
games in extensive form, Γ T . Similar to rationalizability, ISR is a nonequilibrium solution concept, computed by an iterative deletion procedure. The main
difference between ICR and ISR is that the latter is based on sequential rationality: at every history, agents play best responses to their conditional conjectures, and the latter are consistent with Bayesian updating whenever possible.
This distinction is immaterial in static games, where ISR and ICR coincide.
Type space T is nonredundant if ∀ti ti ∈ Ti , ti = ti implies π̂i∗ (ti ) = π̂i∗ (ti ).
Harsanyi’s (1967/1968) definition of Bayesian game does not require the existence of a common prior. The common prior assumption corresponds to the special case that Harsanyi called
consistent.
16
17
640
ANTONIO PENTA
Endogenous Beliefs: Conjectures
At every history, players hold conjectures about their opponents’ behavior,
their types, and the state of nature. These are represented by conditional probability systems (CPS), that is, arrays of conditional beliefs, one for each history.
These beliefs differ from those introduced in Section 2.3 in that they concern
and depend on endogenous variables such as the opponents’ behavior. Hence,
these are endogenous beliefs. To avoid confusion, we thus refer to this kind of
beliefs as conjectures, retaining the term “beliefs” for those introduced in Section 2.3.
For each history h ∈ H, define the event [h] ⊆ Θ0 × T−i × S−i as
[h] = Θ0 × T−i × S−i (h)
(Notice that, by definition, [h] ⊆ [h ] whenever h follows h .)
DEFINITION 2: A conjecture for agent i is a conditional probability system
(CPS hereafter), that is a collection μi = (μi (h))h∈H of conditional distributions μi (h) ∈ Δ(Θ0 × T−i × S−i ) that satisfy the following conditions:
C.1. For all h ∈ H, μi (h) ∈ Δ([h]).
C.2. For every measurable A ⊆ [h] ⊆ [h ], μi (h)[A] · μi (h )[h] = μi (h )[A].
The set of CPS over Θ0 × T−i × S−i is denoted by ΔH (Θ0 × T−i × S−i ).
For each type ti ∈ Ti , his consistent conjectures are
Φi (ti ) = μi ∈ ΔH (Θ0 × T−i × S−i ) : marg μi (φ) = τi (ti ) Θ0 ×T−i
Condition C.1 states that agents are always certain of what they know, that
is, the observed public history18 ; condition C.2 states that agents’ conjectures
are consistent with Bayesian updating whenever possible. Type ti ’s consistent
conjectures agree with his beliefs on the environment at the beginning of the
game.
Sequential Rationality
The set of sequential best responses for type ti to conjectures μi ∈ ΔH (Θ0 ×
T−i × S−i ), denoted by ri (μi |ti ), is defined as
(1)
si ∈ ri (μi |ti ) if and only if ∀h ∈ H(si )
si ∈ arg max
ui (z(si s−i ) θ0 t−i ti ) dμi (h)
si ∈Si (h)
Θ0 ×T−i ×S−i
18
Since players are always certain of what they know (see also discussion in Section 2.2), player
i’s conjectures about θi are omitted: If they were explicitly modelled, condition C.1 would require
that conjectures of payoff type θi satisfy μi (h) ∈ Δ({θi } × [h]) for every h ∈ H.
HIGHER ORDER UNCERTAINTY AND INFORMATION
641
DEFINITION 3: A strategy si ∈ Si is sequentially rational for type ti , written
si ∈ ri (ti ), if there exists μi ∈ Φi (ti ) such that si ∈ ri (μi |ti ).
The notion of sequential rationality is stronger than (normal form) rationality, which only requires that a player optimizes with respect to his initial
conjectures μi (φ). The two notions obviously coincide if the game is static (i.e.
if H = {φ}).
Interim Sequential Rationalizability
Interim sequential rationalizability (ISR) consists of an iterated deletion
procedure for each type of each player. The deletion procedure is described as
follows: for each type ti , reduced strategy si survives the kth round of deletion
if and only if si is sequentially rational for type ti , and if it is justified by conjectures μi that, at the beginning of the game, are concentrated on pairs (t−i s−i )
consistent with the previous rounds of deletion.
DEFINITION 4—ISR: For each i ∈ N, let ISR0i = Ti × Si . Recursively, for
k = 1 2 and ti ∈ Ti , let
ISRk−1
−i =
×
j∈N\{i}
ISRk−1
j
(ti ) : ∃μi ∈ Φi (ti ) s.t. (i) ŝi ∈ ri (μi |ti ) and
ISRki (ti ) = ŝi ∈ ISRk−1
i
(ii) supp(μi (φ)) ⊆ Θ0 × ISRk−1
−i
ISRki = {(ti si ) ∈ Ti × Si : si ∈ ISRi (ti )}
and
ISRk =
×ISR i∈N
Finally, ISR :=
k≥0
k
i
ISRk .
Notice that ISR restricts agents’ conjectures only at the beginning of the
game (condition (ii)). If history h is given zero probability by the conditional
conjectures held at the preceding node, i’s conjectures at h may be concentrated anywhere in Θ0 × T−i × S−i (h). ISR therefore corresponds to the assumption that at the beginning of the game, players commonly believe that
everyone is (sequentially) rational. But if player i observes something unexpected, then he may consider that the opponents are not rational. The fundamental logic of ISR is best understood by considering the case of complete
information first.
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ANTONIO PENTA
FIGURE 1.—Examples 1 and 2.
EXAMPLE 1: Consider the game in Figure 1, and suppose that it is common
knowledge that θ = 0. Denote this model by T CK = {t CK }. Then strategy (In a3 )
is dominated by a1 and deleted at the first round. Strategies a1 and (In a2 ) are
justified by b2 and b1 , respectively. Hence, ISR11 (t1CK ) = {a1 (In a2 )}. Given
this, player 2’s initial conjectures in the second round must put zero probability
on (In a3 ). No further restrictions are imposed. In particular, conjectures μ̂2
can be such that μ̂2 (φ)[(a1 )] = 1 and μ̂2 (In)[(In a3 )] = 1, which makes b2 the
unique sequential best response to μ̂2 . On the other hand, if μ̂2 (φ)[(In a2 )] >
0, Bayesian updating implies that μ̂2 (In)[(In a2 )] = 1, and player 2’s unique
sequential best response is b1 . Given that nothing is deleted for player 2, the
procedure ends here: ISR(t CK ) = {a1 (In a2 )} × {b1 b2 }.
Notice that strategy b2 (and so a1 ) is not deleted here because ISR allows
player 2 to believe, after an unexpected history, that the opponent may play
irrationally (i.e., (In a3 )). In contrast, suppose that we make the following assumption:
[H.1] Even after unexpected moves, player 2 believes that player 1 is rational.
Then, independent of his initial beliefs, in the proper subgame, player 2
always assigns zero probability to (In a3 ) and plays b1 if rational. Hence, if
player 1 believes [H.1] and that player 2 is rational, his unique best response
is (In a2 ). This is the logic of Pearce’s (1984) extensive form rationalizability
(EFR), which delivers ((In a2 ) b1 ) as the unique outcome in this game.
The logic of EFR in Example 1 seems compelling. Yet, as the next example
shows, its predictions are not robust. To illustrate the point, we construct a sequence of hierarchies of beliefs, converging to those implicit in Example 1, in
which (a1 b2 ) is the unique ISR outcome (hence, also the unique EFR outcome). Since that outcome is ruled out by EFR in the limit, but uniquely selected along the converging sequence, EFR is not “robust.” Example 2 also
provides the intuition for the result in Theorem 1.
HIGHER ORDER UNCERTAINTY AND INFORMATION
643
EXAMPLE 2: In the game in Figure 1, let the space of uncertainty be such
that Θ1 = {0 3}, while Θ0 and Θ2 are singletons (hence Θ Θ1 : player 1 is
informed, player 2 is not). Let t ∗ = (t1∗ t2∗ ) represent the situation in which
there is common certainty that θ = 0: t1∗ knows that θ = 0, and puts probability
1 on t2∗ ; t2∗ puts probability 1 on θ = 0 and t1∗ . A reasoning similar to that in
Example 1 implies that {a1 (In a2 )} and {b1 b2 } are the sets of ISR strategies
for t1∗ and t2∗ , written ISR(t ∗ ) = {a1 (In a2 )} × {b1 b2 }.
We construct next a sequence of types {t m }, converging to t ∗ , such that
(a1 b2 ) is the unique ISR outcome for each t m . Since it is the unique ISR
outcome along the sequence, any (strict) refinement that rules it out at t ∗ is
not robust. (Since TΘ∗∗ is endowed with the product topology, the convergence
below is with respect to this topology.)
ε
Fix ε ∈ (0 16 ) and let p ∈ (0 (1−2ε)
). Consider the set of type profiles
ε
ε
ε
∗
3
0
T1 × T2 ⊆ TΘ∗ , where T1 = {−1 1 13 30 33 50 53 } and T2ε = {0 2 4 }.
Types kθ (k = −1 1 3 θ = 0 3) are player 1’s types who know that the
true state is θ. Beliefs are described as follows. Type −13 puts probability 1 on
1
type 0; type 0 assigns probability 1+p
to type −13 , and complementary prob0
3
ability to types 1 and 1 , with weights (1 − ε) and ε, respectively. Similarly,
1
for all k = 2 4 player 2’s type k puts probability 1+p
on player 1’s types
0
3
(k − 1) and (k − 1) , with weights (1 − ε) and ε, respectively, and complep
on the (k + 1) types, with weights (1 − ε) on (k + 1)0
mentary probability 1+p
and ε on (k + 1)3 . For all other types of player 1, with k = 1 3 and θ = 0 3,
1
type kθ puts probability 1+p
on player 2’s type k − 1, and complementary probability on player 2’s type k + 1. (The type space is represented in Figure 2.)
Notice that the increasing sequence of even k’s and odd k0 ’s converges to t ∗ as
we let ε approach 0. It will be shown that player 2’s types 0 2 4 only play
b2 , while player 1’s types 10 30 only play a1 .
Clearly, types k3 (k = −1 1 3 ) play (In a3 ), which they know is domi1
nant. Type 0 puts probability 1+p
on type −1, who plays (In a3 ); given these
initial beliefs, type 0’s conditional conjectures after In must put probability at
FIGURE 2.—The type space in Example 2.
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ANTONIO PENTA
1
1
least 1+p
on (In a3 ), which makes b2 optimal. Type 10 also puts probability 1+p
on type 0, who plays b2 ; thus a1 is the unique best response. Type 2’s initial
conjectures are such that type 10 plays a1 , and types 13 and 33 play (In a3 ).
Hence, the probability of a3 conditional on In must be no smaller than
Pr(θ = 3|not 10 ) =
=
ε
1
(1 − ε)
1−
1+p
(1 + p)ε
p+ε
ε
, this probability is greater than 12 . Playing b2 is thus the
Given that p < (1−2ε)
unique best response, irrespective of type 2’s conjectures about the behavior
of 30 . Given this, type 30 also plays a1 . The reasoning can be iterated, so that
for all types 10 30 50 a1 is the unique ISR strategy, while for all types
0 2 4 of player 2, strategy b2 is.
3.1. Some Remarks on the Solution Concept
ISR generalizes existing solution concepts that have been defined for special
cases. For instance, in games with complete and perfect information, ISR coincides with Ben-Porath’s (1997) common certainty in rationality (CCR). If the
game is static, ISR coincides with ICR. The existing solution concept that is
closest in spirit is Battigalli and Siniscalchi’s (2007) weak Δ-rationalizability,
which is only defined for games without a type space.19 Penta (2010b) also
showed that, in private values environments, ISR is (generically) equivalent to
Dekel and Fudenberg’s (1990) S ∞ W procedure applied to the interim normal
form.20
Like ICR, one important feature of ISR is that agents’ conjectures allow correlation between the opponents’ behavior and the state of nature. Correlated
conjectures are key to the following results from Penta (2010b), useful for the
proof of the structure theorem:
PROPOSITION 1—Proposition 1 in Penta (2010b): ISR is upper hemicontinuous on the universal type space. That is, for each t ∈ TΘ∗ and sequence {t m } : t m → t,
and for {sm } ⊆ S such that sm → ŝ and sm ∈ ISR(t m ) for every m, ŝ ∈ ISR(t).
19
I conjecture that extending an analogous result for ICR in Battigalli, Di Tillio, Grillo, and
Penta (2011), it can be shown that ISR is equivalent to weak Δ-rationalizability where the Δ restrictions are those derived from the type space.
20
The S ∞ W procedure consists of one round of deletion of weakly dominated strategies followed by iterated deletion of strongly dominated strategies. It was introduced by Dekel and Fudenberg (1990), who showed that S ∞ W is “robust” to the possibility that players entertain small
doubts about their opponents’ payoff functions, under the assumption that players know their
own payoffs, which corresponds to the private values case here.
HIGHER ORDER UNCERTAINTY AND INFORMATION
645
PROPOSITION 2—Proposition 2 in Penta (2010b): ISR is type-space-invariant.
Let T and T̃ be two Θ-based type spaces. If ti ∈ Ti t̃i ∈ T̃i are such that π̂ ∗ (ti ) =
∗
π̂ ∗ (t̃i ) ∈ TiΘ
, then ISR(ti ) = ISR(t̃i ). Indeed, for any k, if ti and t̃i are such that
l
l
(ti ) = ISRk−1
(t̃i ).
π̂i (ti ) = π̂i (t̃i ) for all l ≤ k, then ISRk−1
i
i
These results, which can be interpreted as robustness properties (Section 6.1), generalize analogous properties of ICR (Dekel, Fudenberg, and
Morris (2007)).
4. THE STRUCTURE THEOREM FOR ISR
The main result in this section shows that, if all common knowledge assumptions on payoffs are relaxed, ISR is the strongest robust solution concept
among those that satisfy the minimal requirement discussed in the introduction, ICBSR. (Namely, players are sequentially rational and this is common
belief at the beginning of the game.21 )
Since any PI structure entails common knowledge assumptions, investigating
the robustness of solution concepts when no common knowledge assumptions
are imposed essentially means to assume that the underlying space of uncertainty Θ is sufficiently rich. This is the spirit of Weinstein and Yildiz’s richness
condition: for each action of each player, there exists a state θ ∈ Θ in which
that action is strictly dominant. This condition cannot be satisfied in dynamic
games. An analogous condition though can be formulated simply by adopting
a notion of dominance based on sequential rationality.
DEFINITION 5: Strategy si is conditionally dominant at θ ∈ Θ if ∀h ∈ H(si ),
∀si ∈ Si (h), ∀s−i ∈ S−i (h),
si (h) = si (h)
⇒
ui (z(si s−i ) θ) > ui (z(si s−i ) θ)
RICHNESS CONDITION (RC): Let Θ0 (Θi ui )i∈N be such that (i) ∀s ∈ S,
s
∃θs = (θ0s θis θ−i
) ∈ Θ : ∀i ∈ N, si is conditionally dominant at θs and (ii) ∀i ∈ N,
22
Θi is convex.
If the game is static, the notion of conditional dominance coincides with
that of strict dominance. RC therefore coincides with Weinstein and Yildiz’s
(WY) richness condition in static games. Furthermore, in no information environments (i.e., if Θ Θ0 ), RC coincides with Chen’s (2011) “extensive form
21
Penta (2010b) showed that ISR in fact characterizes the behavioral implications of ICBSR.
Condition (ii) in (RC) is a technical assumption, used to perturb payoffs in case of ties between different terminal nodes (see eq. (2), p. 654), which is only required in nongeneric cases.
Condition (ii) can be relaxed if, for instance, payoffs are required to be in generic position for
every θ ∈ Θ.
22
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ANTONIO PENTA
richness.” Hence, the analysis covers both Weinstein and Yildiz’s and Chen’s
environments.23
THEOREM 1: If Θ0 (Θi ui )i∈N satisfies the richness condition (RC), for any
finite type profile t̂ ∈ T̂Θ and any s ∈ ISR(t̂), there exists a sequence of finite type
profiles {t̂ m } ⊆ T̂Θ such that t̂ m → t̂ as m → ∞ and ISR(t̂ m ) = {s} for each m.
Furthermore, for each m, t̂ m belongs to a finite belief-closed subset of types, T m ⊆
TΘ∗ , such that for each m and each t ∈ T m , |ISR(t)| = 1.
Theorem 1 implies that any refinement of ISR (e.g., extensive form rationalizability or sequential equilibrium) is not upper hemicontinuous (see Example 2 in Section 3). To see this, let S be a refinement of ISR (i.e., S (t) ⊆ ISR(t)
for all t). Being a refinement, there exist ti ∈ TΘ∗ and s ∈ S such that s ∈ ISR(t)
and s ∈
/ S (t) By Theorem 1, there exists a sequence {t m } converging to t such
that {s} = ISR(t m ) ⊇ S (t m ), but s ∈
/ S (t). Therefore, S is not upper hemicontinuous. Since ISR is upper hemicontinuous (Proposition 1), the following
statement is true.
COROLLARY 1: ISR is the strongest upper hemicontinuous solution concept
among its refinements.
The proof of Theorem 1 requires a substantial investment in additional concepts and notation, and is relegated to the Appendix. The main points of departure from Weinstein and Yildiz (2007) are due to the necessity of breaking
the ties between strategies at unreached information sets. Although notationally involved, the idea is simple. Consider the sequence constructed in Example 2: to obtain b2 as the unique ISR strategy for player 2, given that player 1
would play a1 , it was necessary to perturb player 2’s beliefs so to assign arbitrarily small probability to types 13 33 who believe that (In a3 ) is dominant.
These “dominance types” play the role of trembles and allow one to break the
tie between b1 and b2 .
It is proved next that under the richness condition, ISR is generically unique
on the universal type space. The proof exploits the following known result.
LEMMA 1—Mertens and Zamir (1985): The set T̂Θ of finite types is dense in
TΘ∗ , that is,
TΘ∗ = cl(T̂Θ )
Since both Weinstein and Yildiz (2007) and Chen (2011) implicitly assumed |Θi | = 1 for all
i ∈ N, condition (ii) is vacuously satisfied in their settings. RC thus formally encompasses both
settings.
23
HIGHER ORDER UNCERTAINTY AND INFORMATION
647
THEOREM 2: Under the richness condition (RC), the set
U = {t ∈ TΘ∗ : |ISR(t)| = 1}
is open and dense in TΘ∗ . Moreover, the unique ISR outcome is locally constant, in
the sense that ∀t ∈ U such that ISR(t) = {s}, there exists an open neighborhood of
types, Nδ (t), such that ISR(t ) = {s} for all t ∈ Nδ (t).
PROOF: U is dense. To show that U is dense, notice that by Proposition 2, for
any t̂ ∈ T̂ there exists a sequence {t̂ m } ⊆ T̂ m such that t̂ m → t̂ and ISR(t̂ m ) = {s}
for some s ∈ ISR(t̂). By definition, t̂ m ∈ U for each m. Hence, t̂ ∈ cl(U ), thus
T̂ ⊆ cl(U ). But we know that cl(T̂ ) = T ∗ ; therefore, cl(U ) ⊇ cl(T̂ ) = T ∗ . Hence
U is dense.
U is open and ISR locally constant in U . Since (Proposition 1) ISR is upper
hemicontinuous, for each t ∈ U there exists a neighborhood Nδ (t) such that for
each t ∈ Nδ (t), ISR(t ) ⊆ ISR(t). Since ISR(t) = {s} for some s and ISR(t ) =
∅, it follows trivially that ISR(t ) = {s}, hence Nδ (t) ⊆ U . Therefore, U is open.
By the same token, we also have that ISR(t ) = {s} for all t ∈ Nδ (t), that is, the
unique ISR outcome is locally constant.
Q.E.D.
COROLLARY 2: Generic uniqueness of ISR implies generic uniqueness of any
equilibrium refinement (in particular, of any perfect-Bayesian equilibrium outcome).
For each s ∈ S, let U s = {t ∈ T̂Θ : ISR(t) = {s}}. From Theorem 2, we know
that these sets are open. Let the boundary be bd(U s ) = cl(U s ) \ U s .
COROLLARY 3: Under the richness condition, for each t ∈ T̂Θ , |ISR(t)| > 1 if
and only if there exist s s ∈ ISR(t) : s = s such that t ∈ bd(U s ) ∩ bd(U s ).
Theorems 1 and 2 together imply a structure theorem for ISR analogous to
the one that Weinstein and Yildiz proved, under the no information assumption, for ICR: ISR is a generically unique and locally constant solution concept
that yields multiple solutions at, and only at, the boundaries where the concept
changes its prescribed behavior.
Since ISR coincides with ICR in static games, Theorems 1 and 2 extend Weinstein and Yildiz’s results to arbitrary information structures.24
24
Given the results above, analogues of the remaining results in Weinstein and Yildiz (2007)
can be obtained in a straightforward manner for ISR: in particular, it can be shown that Theorem 1 also holds if one imposes the common prior assumption.
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ANTONIO PENTA
FIGURE 3.—Examples 3 and 4.
4.1. Extensive Form versus Normal Form Approach
Independent work by Chen (2011) considered the robustness question in
dynamic games. Chen showed that the structure theorem for ICR holds in dynamic games, once the richness condition is suitably adapted.25 There are two
main differences between Chen’s and the approach followed in this paper: first,
Chen only considers the case of no information; second, he maintains a normal
form approach, that is, he applies ICR to the reduced normal form of the game.
These two points are closely related: if no common knowledge restrictions on
payoffs are imposed, the no information assumption implies that agents’ beliefs on payoffs at unexpected histories are unrestricted. Sequential rationality
therefore has no bite at unexpected histories and coincides with normal form
rationality.
EXAMPLE 3: Consider the game in Figure 3, and assume that Θ = {3} (complete information). Denote by t CK the degenerate hierarchy in which θ = 3
is common certainty. Normal form rationalizability (or ICR) does not rule
out anything in this game, while ISR delivers the backward induction solution
(Out D).
Now, suppose instead that θ ∈ {3 −3} and that agents share common certainty of θ = 3, but have no information (hence, Θ∗ Θ∗0 = {−3 3}). Denote
by (t1A t2A ) ∈ TΘ∗ ∗ players’ hierarchies that correspond to (initial) common certainty of θ = 3. If player 1 believes that θ = 3 and that player 2 will play D,
player 1’s optimal response is to play Out. If player 2 believes this, his information set is unexpected and ISR allows him to revise his beliefs in favor
of θ = −3. In this case, player 2’s best response is U. If player 1 anticipates
this, then playing In is optimal even if she believes that θ = 3. That is because
player 1 knows that player 2, although initially certain of θ = 3, does not know
that θ = 3. Hence, ISR(t A ) coincides with ICR(t A ) in this example. If players
have no information on payoffs, not even one round of backward induction
reasoning is robust. In this sense, the very notion of dynamic game is immaterial without information.
25
zon.
Recently, Weinstein and Yildiz (2010) extended Chen’s analysis to games with infinite hori-
HIGHER ORDER UNCERTAINTY AND INFORMATION
649
This insight has general validity.
COROLLARY 4: If the PI structure has no information and the richness condition is satisfied, ISR and ICR coincide everywhere on the universal type space.26
One possible interpretation of Corollary 4 is that, for what concerns the impact of higher order uncertainty, there is no difference between static and dynamic games. It is important though to emphasize that this is the case under
the no information assumption, precisely because the very notion of sequential rationality has no bite in these settings.27 Under alternative information
structures, the normal form approach is inadequate to study the robustness of
solution concepts based on sequential rationality (e.g., subgame perfect or sequential equilibrium), which are the main points of interest in dynamic games.
There is therefore an intimate connection between agents’ information and the
extensive form approach adopted here.
5. INVARIANCE WITH RESPECT TO INFORMATION
Consider the following example.
EXAMPLE 4: As in Example 3, let Θ∗ = {−3 3}, but this time assume that
player 2 observes the realization of θ and that this is common knowledge
(Θ∗ Θ∗2 = {3 −3}). Consider again the case in which players share initial
common certainty of θ = 3, denoted by t B = (t1B t2B ): type t2B knows that θ = 3
and puts probability 1 on t1B ; t1B puts probability 1 on θ = 3 and t2B . Since player 1
knows that player 2 knows the true state θ, if player 1 is rational and believes
that player 2 is rational, he must play Out: if player 2 is rational and knows
θ, player 1 obtains −3 in the subgame irrespective of the realization of θ. The
proper subgame at this point is unexpected, but type t2B knows that θ = 3, therefore he plays D. In this case, ISR(t B ) coincides with the backward induction
solution of the complete information model.
Similar to types t A from Example 3, types t B share the same hierarchy as
types t CK . Yet, while ISR(t A ) = ICR(t CK ) ⊃ ISR(t CK ), we have that ISR(t B ) =
ISR(t CK ) ⊂ ICR(t CK ). In both examples, the complete information model t CK
was embedded in a richer PI structure, Θ∗ , and envisioned as the common certainty types t A and t B , respectively. Only in the second example though could
26
Theorem 3 in Chen (2011) obtained this result from the structure theorem for ICR and
upper hemicontinuity of ISR (Proposition 1). A simple direct proof is also possible, following the
argument sketched at the beginning of this section.
27
In fact, it can be shown that Corollary 4 holds whenever players do not have “enough” infors
mation about payoffs. Formally, if for every i and every θi ∈ Θi , ∀s ∈ S, ∃(θ0s θ−i
) such that si is
s
s
conditionally dominant at (θ0 θi θ−i ), then ICR and ISR coincide everywhere on the universal
type space.
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ANTONIO PENTA
this be done without affecting the predictions of ISR. This suggests a novel
notion of invariance.
DEFINITION 6: PI structure Θ0 (Θi ui )i∈N is embedded in PI structure
Θ∗0 (Θ∗i u∗i )i∈N if Θk ⊆ Θ∗k for each k = 0 1 n and u∗i (z θ) = ui (z θ)
for all (z θ i) ∈ Z × Θ × N.
In Example 3, the complete information model Θ = {3} was embedded in
the PI structure Θ∗ Θ∗0 = {3 −3}, while in Example 4 it was embedded in the
PI structure Θ∗ Θ∗2 = {3 −3}.
Given a PI structure Θ0 (Θi ui )i∈N , let ti ∈ TiΘ denote a type in some Θbased type space. Any such type induces a Θ-hierarchy π̂i∗ (ti ) = (θi (ti ) π̂i1 (ti )
∗
. Now consider a richer PI structure Θ∗0 (Θ∗i u∗i )i∈N that embeds
) ∈ TiΘ
Θ0 (Θi ui )i∈N . Since Θk ⊆ Θ∗k for all k, π̂i∗ (ti ) can be naturally embedded in
the Θ∗ -based universal type space and can be seen as a Θ∗ -based hierarchy. Let
∗
∗
∗
∗
→ TiΘ
βi : TiΘ
∗ denote such embedding and let κi ≡ βi ◦ π̂i .
A
In Examples 3 and 4, the common certainty types t and t B have the same
∗
hierarchy as the common knowledge type t CK (e.g., κ∗i (tiCK ) = tiA ∈ TiΘ
∗ ).
DEFINITION 7: A solution concept Si : Ti ⇒ Si is information-invariant if, for
any Θ0 (Θi ui )i∈N and Θ0 (Θi ui )i∈N embedded in Θ∗0 (Θ∗i u∗i )i∈N , and
∗
∗ for any ti ∈ TiΘ and ti ∈ TiΘ
, if κi (ti ) = κi (ti ), then Si (ti ) = Si (ti ).
As shown by Example 3, ISR is not information-invariant in general, while
it satisfies information invariance in Example 4. The reason for such different
behavior is that under the no information assumption, moving to a PI structure
with a richer set of states entails more freedom to specify a player’s beliefs
about his own payoffs, thereby changing the set of sequential best responses.
In contrast, in Example 4, player 2 knows his own payoffs (and this is common
knowledge). Hence, even if ISR leaves players’ beliefs at unexpected histories
unrestricted, a type’s preferences over the terminal nodes do not change. This
provides the intuition for the information-invariance result in private values
settings.
PROPOSITION 3: In static games, ISR is information-invariant. In dynamic
games, ISR is information-invariant in environments with private values.
For the proof, see Appendix B.
The first part of Proposition 3 follows trivially from the fact that the assumptions on information only play a role at unexpected histories. The driving force
behind the second part of the proposition is that in environments with private
values, players know their payoffs and therefore their beliefs about them never
HIGHER ORDER UNCERTAINTY AND INFORMATION
651
change.28 It is only such belief stability that really matters for the invariance
result. (See the discussion in Section 6.2.)
It is easy to see that information invariance holds also outside of private values (e.g., Example 4 is not with private values). For instance, it can be shown
that Proposition 3 holds provided that players have enough information on
their preference ranking over the terminal histories. That is, if the PI struc
)∈
tures are such that for every i and every θi ∈ Θi , for any (θ0 θ−i ) (θ0 θ−i
Θ0 × Θ−i and any z z ∈ Z , ui (z θi (θ0 θ−i )) > ui (z θi (θ0 θ−i )) if and only
)) > ui (z θi (θ0 θ−i
)).
if ui (z θi (θ0 θ−i
6. DISCUSSION
6.1. Robustness(-es)
If we let Θ∗ denote a space of uncertainty that satisfies the richness condition
and we let TΘ∗∗ denote the Θ∗ -based universal type space, assuming common
knowledge of TΘ∗∗ entails no essential loss of generality. TΘ∗∗ can therefore be
thought of as a universal model and can be used to investigate the robustness
of game theoretic predictions “when all common knowledge assumptions are
relaxed.” The continuity of a solution concept on such a universal model therefore corresponds to a specific robustness property: robustness with respect to
small “mistakes” in the modelling choice of which subset of players’ hierarchies to consider. Proposition 1 states that ISR is robust in this sense, under all
information structures. Furthermore, Theorem 1 implies that no refinement of
ISR is robust in this sense.
In modelling a strategic situation, when a subset of Θ∗ -hierarchies is selected, it is common to represent them by means of (nonuniversal) Θ∗ -based
type spaces, TΘ∗ (Definition 1). This modelling practice does not change the
common knowledge assumptions on the PI structure, but imposes restrictions
on beliefs that entail some loss in generality. A solution concept is type-spaceinvariant if the behavior prescribed for a given hierarchy does not depend on
whether it is represented as an element of TΘ∗∗ or as a type in a (nonuniversal)
Θ∗ -based type space. Proposition 2 therefore states another robustness property for ISR. This one also holds irrespective of the information structure.
We typically make common knowledge assumptions not only on players’ beliefs, but also on payoffs. For instance, if all types in TΘ∗ have beliefs concentrated on some strict subset Θ ⊂ Θ∗ , that is, there is common certainty of Θ in
TΘ∗ , it is common to exclude from the model states in Θ∗ \ Θ. This way, common certainty of Θ is turned into common knowledge of Θ, and hierarchies are
represented by Θ-based type spaces, where Θ is a PI structure embedded in Θ∗
28
Such “beliefs stability” is implicit in our setup, because i’s beliefs about θi are not modelled
explicitly. But, as discussed in Section 2.2, explicitly modelling such beliefs would be redundant,
because knowledge implies beliefs stability.
652
ANTONIO PENTA
(Definition 6). A solution concept is information-invariant (Definition 7) if its
predictions are robust to the introduction of these extra common knowledge
assumptions. Information invariance is therefore yet another robustness property, not previously considered by the literature.29 As discussed in Section 5,
ISR is information-invariant in environments with private values, not in environments with no information.
6.2. A Subjectivist View: Strong Belief
The setting of this paper rests on two assumptions:
A.1. Every player knows his own payoff type in every state.
A.2. A player with payoff types θ̂i never abandons the belief that θi = θ̂i
(while he may abandon beliefs about (θ0 θ−i )).
As discussed in Section 2.2, within the classical “possible worlds” models,
condition A.2 is a natural consequence of the requirement (implicit in standard
models in information economics) that agents’ beliefs do not contradict their
information.
An alternative interpretation of payoff states is also possible and does not
require the notion of an external true state. Under this interpretation, A.2 does
not follow naturally from A.1, and it may be interesting to look at the two
assumptions separately. In this alternative “subjectivist” view, payoff types θ̂i
represent those beliefs about payoffs that player i is never willing to abandon
(his strong beliefs). A payoff state no longer represents information about the
true payoffs anymore, but a realization of individuals’ epistemic states. None
of the results above is affected by this alternative interpretation. I maintain the
terminology of the “possible worlds” model simply because it is the standard
paradigm in information economics.30
From this perspective, it is easy to see how a proper relabelling of the model
can accommodate several variations of A.2. For instance, relaxing A.2 in a
given environment is always equivalent to considering some other environment with no information. Consider the following example: Let N = {1 2},
Θ0 := Θ10 × Θ20 = [1 2]2 , Θi = [0 3], and ui (z θ) = (θi − θ0i )Ui (z) for some
Ui (·) ∈ [0 1]Z . Player i observes θi , but not θ0i . To relax A.2, we can reformulate the model, letting Θ̂0 = Θ̂10 × Θ̂20 = [−2 2]2 and ûi (z θ) = θ̂0i · Ui (z) for
each i = 1 2. Similar transformations can also accommodate some strengthened versions of A.2, in which players entertain strong beliefs about components other than θi . Suppose, for instance, that player i is never willing to
change his beliefs about θ0i . Then the environment can be rewritten as one
29
Dekel, Fudenberg, and Morris (2007) studied the upper hemicontinuity and type space invariance of ICR.
30
The subjectivist view is well known in other fields such as philosophy and computer science.
For instance, the classic textbook by Gardenfors (1988) analyzes the logic of epistemic change
without any reference to an external state and the corresponding notions of truth and falsehood.
HIGHER ORDER UNCERTAINTY AND INFORMATION
653
of private values, setting Θ̂i = [−2 2] and ûi (z θ) = θ̂i · Ui (z) for each i = 1 2.
Due to the restrictiveness of the information partition considered here (with a
product structure), not all strong beliefs can be accommodated this way. (For
instance, if θ01 and θ02 in the example above are replaced by a common value
component θ0 , then the product structure cannot accommodate strong belief
in θ0 .) However, a similar relabelling is always possible if general information
partitions over Θ are considered. This generalization seems to be conceptually
straightforward, so I leave it to future research.
APPENDIX
A. Proving Theorem 1
The proof exploits a refinement of ISR—strict sequential rationalizability
(SSR)—in which strategies that are never strict sequential best responses are
deleted at each round. The proof involves two main steps: in the first step
(Lemma 3), it is shown that if si ∈ ISRi (ti ) for finite type ti , then si is also
SSR for some nearby type ti ; in the second step (Lemma 4), it is shown that
by perturbing beliefs further, any si ∈ SSRi (ti ) can be made uniquely ISR for a
type close to ti .
DEFINITION 8: Fix a Θ-based type space, T = Θ0 (Θi Ti θi τi )i∈N . Let
SSR0i = Ti × Si . Recursively, for each k = 1 2 and ti ∈ Ti , let
SSRk−1
−i =
× SSR
j=i0
k−1
j
(ti ) : ∃μi ∈ Φi (ti ) s.t.
SSRki (ti ) = ŝi ∈ SSRk−1
i
(i) ri (μi |ti ) = {ŝi } (ii) supp(μi (φ)) ⊆ Θ∗0 × SSRk−1
−i
k−1
i
(iii) if t−i ∈ supp marg μ (φ) and s−i ∈ SSR−i (t−i )
∗
T−iΘ
then: s−i ∈ supp marg μi (φ) S−i
SSRki = {(ti si ) ∈ Ti × Si : si ∈ SSRki (ti )}
and
SSRk =
×SSR i∈N
Finally, SSR =
k
i
SSRk .
k≥0
The following lemma states a standard fixed-point property for SSR.
654
ANTONIO PENTA
LEMMA 2: Let {Vj }j∈N be such that for each i ∈ N, Vi ⊆ Si ×Ti , and ∀si ∈ Vi (ti ),
∃μi ∈ Φi (ti ),
(i) supp(μi (φ)) ⊆ j=i Vj ,
(ii) {si } = ri (μi |ti ).
Then Vi (ti ) ⊆ SSRi (ti ).
×
Exploiting the richness condition, let Θ̄ ⊂ Θ be a finite set of dominance
states, such that ∀s ∈ S, ∃!θs ∈ Θ̄ at which s is conditionally dominant. For
s
] = 1.
each s ∈ S, let t̄ s ∈ TΘ∗ be such that for each i, θi (t̄ s ) = θis and τi (t̄is )[θ0s t̄−i
s
Let T̄ = {t̄ : s ∈ S}, and let T̄j and T̄−j denote the corresponding projections.
Elements of T̄i will be referred to as dominance types, and will play the role of
the ka3 types in Example 2.
For each i and si ∈ Si , let T̄−i (si ) be such that ∀s−i ∈ S−i ∃!t̄−i ∈ T̄−i (si ) such
(s s )
that t̄−i = t̄−ii −i . Notice that for each t̄is ∈ T̄i , {si } = SSR1i (t̄is ), because si is the
unique sequential best reply to any conjecture consistent with condition (iii) in
Definition 8.
LEMMA 3: Under the richness condition, for any finite type ti ∈ T̂iΘ , for any
si ∈ ISRi (ti ), there exists a sequence of finite types {ιm (ti si )}m∈N , such that the
following statements hold:
(i) ιm (ti si ) → ti as m → ∞.
(ii) ∀m, si ∈ SSRi (ιm (ti si )) and ιm (ti si ) ∈ T̂iΘ .
(iii) ∀m, conjectures μsi m ∈ Φ(ιm (ti si )) such that {si } = ri (μsi m |ιm (ti si ))
satisfy
T̄−i (si ) ⊆ supp marg μsi m (φ) ∗
T−iΘ
PROOF: Step I: Fix ti ∈ T̂iΘ . For each k = i, let Θk be the finite set of payoff
states that receive positive probability by ti . For each si ∈ ISR(ti ), ∃μsi ∈ Φi (ti )
such that (i) si ∈ ri (μsi |ti ) and (ii) supp(μsi (φ)) ⊆ Θ0 × ISR−i . Given a probability space (Ω B ) and a set A ∈ B , denote by υ[A] the uniform probability
distribution concentrated on A. For each ε ∈ [0 1], consider the set of types
profiles i∈N Tiε ⊆ TΘ∗ such that each Tiε consists of all the types t̄i ∈ T̄i and of
types ῑi (ti si ε) such that
×
(2)
θi (ῑi (ti si ε)) = εθis + (1 − ε)θi (ti )
and
(3)
τiε (ῑi (ti si ε)) = ευ[{θ0s }×T̄−i (si )] + (1 − ε)[μsi (φ) ◦ ι̂−1
−iε ]
ε
is the subset of dominance type profiles defined above, and
where T̄−i ⊆ T−i
ε
ι̂−iε : Θ0 × T−i × S−i → Θ0 × T−i
HIGHER ORDER UNCERTAINTY AND INFORMATION
655
is such that
ι̂−iε (θ0 s−i t−i ) = (θ0 ῑ−i (t−i s−i ε))
By construction, with probability ε, type ῑi (ti si ε) is certain that si is conditionally dominant and puts positive probability on all of the opponents’ domiε
ε
→ Θ0 × T−i
× S−i such that
nance types in T̄−i . Define γ : Θ0 × T−i
ε
\ T̄−i for every ῑ−i (t−i s−i ε) ∈ T−i
γ(θ0 ῑ−i (t−i s−i ε)) = (θ0 ῑ−i (t−i s−i ε) s−i )
and for every
s
ε
∈ T̄−i ⊆ T−i
t̄−i
s
s
γ(θ0 t̄−i
) = (θ0 t̄−i
s−i )
ε
Consider the conjectures μ̂i ∈ ΔH (Θ0 × T−i
× S−i ) defined by
ε
μ̂i (φ) = τiε (ῑi (ti si ε)) ◦ γ −1 ∈ Δ(Θ0 × T−i
× S−i )
For any ε > 0, the conjectures μ̂i are such that T̄−i ⊆ supp(margT ε μ̂i (φ)).
−i
From the definition of γ, it follows that supp(margS−i μ̂i (φ)) = S−i , so that the
entire CPS (μ̂i (h))h∈H can be obtained via Bayes’ rule. This also implies that μ̂i
satisfies condition (iii) in the definition of SSR. Furthermore, by construction,
μ̂i ∈ Φi (ῑi (ti si ε)), and ∀ε > 0, ∀h ∈ H, ∃ηεh ∈ (0 1) such that ηεh → 0 as
ε → 0 and
marg μ̂i (h) = ηεh · marg v[{θs }×T̄−ih (si )×S−i ]
Θ0 ×Θ−i ×S−i
Θ0 ×Θ−i ×S−i
+ (1 − ηεh ) ·
(s s
)
0
marg μsi (h)
Θ0 ×Θ−i ×S−i
h
where T̄−i
(si ) = {t̄−ii −i : s−i ∈ S−i (h)}. Hence, the conditional conjectures μ̂i (h)
of type ῑ−i (t−i s−i ε) are a mixture: with probability (1 − ηεh ), they agree with
μsi (φ), which made si the sequential best response; with probability ηεh , they
s
h
(si )}, which together
are concentrated on payoff states {θ0i } × {θ−i (t−i ) : t−i ∈ T̄−i
si
with the fact that θi (ῑi (ti si ε)) = εθi + (1 − ε)θi (ti ) breaks all the ties in favor
of si , so that ri (μ̂i |ῑi (ti si ε)) = {si }. Thus, si ∈ SSRi (ῑi (ti si ε)), so that (ii) and
(iii) in the lemma are satisfied for all ε > 0.
The remainder of the proof guarantees that part (i) in the lemma also holds
and it is identical to WY’s counterpart.
Step II. We show that π̂i∗ (ῑi (ti si ε)) → π̂i∗ (ti ) as ε → 0. By construction,
τi (ῑi (ti si ε)) are continuous in ε, hence π̂i∗ (ῑi (ti si ε)) → π̂i∗ (ῑi (ti si 0)) as
ε → 0. It suffices to show that π̂i∗ (ῑi (ti si 0)) = π̂i∗ (ti ) for each ti and i. This is
proved by induction. The payoff types and the first-order beliefs are the same.
656
ANTONIO PENTA
k−1
l
For the inductive step, assume that (π̂il (ῑi (ti si 0)))k−1
l=0 = (π̂i (ti ))l=0 . We show
k−1
k−1
k
k
l
that π̂i (ῑi (ti si 0)) = π̂i (ti ). Define D−i = {(π̂−i (t−i ))l=0 : t−i ∈ T−i }. Under
the inductive hypothesis, it can be shown (see WY) that
(4)
i
marg [μi (φ) ◦ ι̂−1
−i ] = marg μ (φ)
Θ0 ×Dk−1
−i
Θ0 ×Dk−1
−i
Therefore,
π̂ik (ῑi (ti si 0)) = υ[π̂ k−1 (ῑi (ti si 0))] × marg [μsi (φ) ◦ ι̂−1
−i ]
−i
Θ0 ×Dk−1
−i
= υ[π̂ k−1 (ῑi (ti si 0))] × marg μsi (φ)
−i
Θ0 ×Dk−1
−i
= υ[π̂ k−1 (t−i )] × marg μsi (φ)
−i
Θ0 ×Dk−1
−i
= υ[π̂ k−1 (t−i )] × marg τi (ti ) = π̂ik (ti )
−i
Θ0 ×Dk−1
−i
where the first equality is the definition of kth level belief, the second equality is from (4), the third equality is from the inductive hypothesis, the fourth
inequality is from the fact that μsi ∈ Φ(ti ), and the last inequality is again by
definition.
Q.E.D.
LEMMA 4: Under the richness condition, for each finite type t̂i ∈ T̂iΘ , for each
k, for each si ∈ SSRki (t̂i ) such that the conjectures μsi ∈ Φ(t̂i ) : {si } = ri (μsi |t̂i )
satisfy
T̄−i (si ) ⊆ supp marg μsi (φ) ∗
T−iΘ
there exists t̃i ∈ T̂i such that the following statements hold:
(i) For each k ≤ k, π̂ k (t̂i ) = π̂ k (t̃i ).
(ii) ISRk+1
(t̃i ) = {si }.
i
t̃
t̃
(iii) t̃i ∈ T̃i i for some finite belief-closed set of types T̃ t˜i = j∈N T̃j i such that
|ISRk+1 (t)| = 1 for each t ∈ T̃ t̃i .
Hence, for any such si ∈ SSRi (t̂i ) there exists a sequence of finite types tim → t̂i
such that ISRi (tim ) = {si }.
×
PROOF: The proof is by induction. For k = 0, let t̃i be such that θi (t̃i ) = θi (t̂i )
and τi (t̃i ) = υ[{θs }×T̄−i (si )] . Clearly, ISR1i (t̃i ) = {si } and condition (i) is satisk
(t−i )}k−1
fied. Fix k > 0, and write each t−i = (λ κ), where λ = {π̂−i
k =0 and κ =
k−1
∞
k
∗
{π̂−i (t−i )}k =k . Let L−i = {λ : ∃κ s.t. (λ κ) ∈ T−i }. As the inductive hypothesis,
HIGHER ORDER UNCERTAINTY AND INFORMATION
657
assume that for each finite t−i = (λ κ) and s−i ∈ SSRk−1
−i (t−i ) such that T̄−i (si ) ⊆
s
s
supp(margT̂−i μsi (φ)), there exists finite t−i−i = (λ κs−i λ ) such that ISRki (ti i ) =
{si }. Taking any si ∈ SSRki (t̂i ) such that T̄−i (si ) ⊆ supp(margT̂−i μsi (φ)), we con
(t̃i ) = {si }.
struct a type t̃i such that for each k ≤ k, π̂ik (t̂i ) = π̂ik (t̃i ) and ISRk+1
i
∗
By definition, if si ∈ SSRki (t̂i ), ∃μsi ∈ ΔH (Θ0 × T−iΘ
× S−i ) such that
τi (t̂i ) = marg μsi (φ)
Θ×T−i
supp(μsi (φ)) ⊆ Θ0 × SSRk−1
−i {si } = ri (μsi |t̂i )
Using the inductive hypothesis, define the mapping
∗
supp
marg μsi (h) → Θ0 × T−iΘ
ϕ:
Θ0 ×Lk−1
−i ×S−i
h∈H
ϕ(θ λ s−i ) = (θ (λ κs−i λ ))
s.t.
Define type t̃i as
τi (t̃i ) =
marg
μsi (·|φ) ◦ ϕ−1
Θ0 ×Lk−1
−i ×S−i
= μsi (φ) ◦ proj−1
◦ ϕ−1 Θ ×Lk−1 ×S
−i
−i
0
By construction (for the inductive hypothesis), the first k orders of beliefs are
the same for ti and t̃i (which is point (i) in the lemma):
π̃ik (t̃i ) = marg τi (t̃i )
Θ0 ×Lk−1
−i
◦ ϕ−1 ◦ proj−1
= μsi (φ) ◦ proj−1
Θ ×Lk−1 ×S
Θ ×Lk−1
0
−i
−i
0
−i
−1
Θ0 ×Lk−1
−i
= μ (φ) ◦ proj
= μsi (φ) ◦ proj−1
◦ proj−1
Θ0 ×T ∗
Θ ×Lk−1
si
−iΘ
0
−i
= marg τi (ti ) = π̃ (ti )
Θ0 ×Lk−1
−i
k
i
The first equality is by definition, the second equality is from construction of
τi (t̃i ), and the third equality is from the definition of ϕ, for which
◦ ϕ−1 ◦ proj−1
= proj−1
proj−1
Θ ×Lk−1 ×S
Θ ×Lk−1
Θ ×Lk−1
0
−i
−i
0
−i
0
−i
658
ANTONIO PENTA
The fourth and fifth equalities are notational, and the last equality is by def(t̃i ) = {si }. To this end, notice that each
inition. We need to show that ISRk+1
i
(θ0 t−i ) ∈ supp(τi (t̃i )) is of the form (θ0 t−i ) = (θ0 (λ κs−i λ )) and it is such
that ISRk−i ((λ κs−i λ )) = {s−i }. Hence, the CPS consistent with t̃i and with the
restrictions of ISRk−i are μ̃i such that
τi (t̃i ) = marg μ̃i (φ)
∗
Θ∗0 ×T−iΘ
and
μ̃i (φ) Θ0 × {(t−i s−i ) : ISRk−i (t−i ) = {s−i }} = 1
Since T̄−i (si ) ⊆ supp(margT ∗ μsi ) by hypothesis, we have that (from the def−iΘ
inition of T̄−i (si )) t−i ∈T̄−i (si ) ISRk−i (t−i ) = S−i . Hence the conditional conjectures are uniquely determined for all h ∈ H. These conjectures are given by
μ̃i (φ) = τi (t̃i ) ◦ κ −1 , with κ defined as
κ(θ0 (λ κs−i λ )) = (θ0 (λ κs−i λ ) s−i )
Furthermore, for each h,
marg μ̃i (h) = marg μsi (h)
Θ×S−i
Θ×S−i
To see this, given the observation that supp(margS−i μsi (φ)) = S−i , it suffices
to show that margΘ×S−i μ̃i (φ) = margΘ×S−i μsi (φ). But this is immediate, given
that from the definition of κ and ϕ, we have
proj−1
◦ κ ◦ ϕ = I
Θ ×Lk−1 ×S
0
−i
−i
(I is the identity map.) Hence, μ̃i is uniquely determined for all h and it is
equal to μsi , which makes si the unique best response. Hence ISRk+1
(t̃i ) = {si }.
i
The proof of statement (iii) in the lemma is identical to WY’s: Define
s−i
t−i
t̃i
Ti
T̃i = {t̃i } ∪
s
−i
(θt−i
)∈supp(τi (t̂i ))
t̃
T̃j i =
s
t −i
T−i−i
for j = i
s−i
(θt−i
)∈supp(τi (t̂i ))
Given the lemmata above, the proof of Theorem 1 is immediate:
Q.E.D.
659
HIGHER ORDER UNCERTAINTY AND INFORMATION
PROOF OF THEOREM 1: Take any t̂ ∈ T̂ and any s ∈ ISR(t̂). For each i, from
Lemma 3 there exists a sequence {tim } ⊆ T̂iΘ of finite types such that tim → t̂i
and for each i, si ∈ SSRi (tim ) for each m, for conjectures μsi as in the thesis of
Lemma 3 and in the hypothesis of Lemma 4. Then we can apply Lemma 4 to
the types tim for each m: for si ∈ SSRi (tim ), for each k, there exists a sequence
{t̃imk }k∈N such that t̃imk → tim for k → ∞ such that ISRi (t̃imk ) = {si }. Because
the universal type space is metrizable, there exists a sequence km → ∞ with
timkm → t̂i . Set t̂im = timkm , so that t̂ m → t̂ as m → ∞ and ISR(t̂ m ) = {s} for
each m.
Q.E.D.
B. Proof of Proposition 3
The first part follows trivially from the fact that the assumptions on information only play a role at unexpected histories. For the second part, it suffices to show that for any Θ0 (Θi ui )i∈N embedded in Θ∗0 (Θ∗i u∗i )i∈N , for
T ∗∗
T
any TΘ and for any ti ∈ TiΘ , ISRi Θ (ti ) = ISRi Θ (κ∗i (ti )). The proof is by inT ∗∗ 1
T 1
duction. Let ti∗ = κ∗i (ti ). Clearly, if θi (ti ) = π̂i0 (ti∗ ), ISRi Θ (ti ) = ISRi Θ (ti∗ ).
n ∗ k−1
As the inductive hypothesis, assume that (π̂in (ti ))k−1
n=0 = (π̂i (ti ))n=0 implies
TΘ∗∗ k
TΘ k
(ti∗ ), and suppose that (π̂in (ti ))kn=0 = (π̂in (ti∗ ))kn=0 . We
that ISRi (ti ) = ISRi
T ∗∗ k+1
T k+1
show that (π̂in (ti ))kn=0 = (π̂in (ti∗ ))kn=0 implies that ISRi Θ (ti ) = ISRi Θ
(ti∗ ).
TΘ k
Under the inductive hypothesis, s−i ∈ ISRi (t−i ) for some t−i ∈ supp τi (ti )
T ∗∗ k
∗
∗
if and only if s−i ∈ ISR−iΘ (t−i
) for some t−i
∈ supp τi (ti∗ ). In private values
environments, only the conjectures about S−i are payoff relevant for player
i (θ−i ’s do not affect i’s payoffs, and Θ0 is a singleton). Thus, under the inT k+1
ductive hypothesis, ∃μi ∈ Φi (ti ) such that supp(margT−iΘ ×S−i μi ) ⊆ ISR−iΘ
and si ∈ ri (μi |ti ) if and only if ∃μ̂i ∈ Φi (ti ) such that supp(margT ∗ ∗ ×S−i μ̂i ) ⊆
−iΘ
T ∗∗ k+1
and si ∈ ri (μ̂i |ti∗ ). (Remember, the only restrictions on the conISR−iΘ
jectures over S−i imposed by ISR are at the beginning of the game.) Hence
T ∗∗ k+1
T k+1
ISRi Θ (ti ) = ISRi Θ
(ti∗ ).
Q.E.D.
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Manuscript received March, 2010; final revision received May, 2011.